Transverse Linearization for Controlled Mechanical Systems With Several Passive Degrees of Freedom

Abstract

This study examines the mechanical systems with an arbitrary number of passive (non-actuated) degrees of freedom and proposes an analytical method for computing coefficients of a linear controlled system, solutions of which approximate dynamics transverse to a feasible motion. This constructive procedure is based on a particular choice of coordinates and allows explicit introduction of a moving Poincaré section associated with a nontrivial finite-time or periodic motion. In these coordinates, transverse dynamics admits analytical linearization before any control design. If the forced motion of an underactuated mechanical system is periodic, then this linearization is an indispensable and constructive tool for stabilizing the cycle and for analyzing its orbital (in)stability. The technique is illustrated with two challenging examples. The first one is stabilization of a circular motions of a spherical pendulum on a puck around its upright equilibrium. The other one is creating stable synchronous oscillations of an arbitrary number of planar pendula on carts around their unstable equilibria.Index Terms-Moving Poincaré section, orbital stability, synchronisation of mechanical system, transverse linearization, underactuated mechanical systems, virtual holonomic constraints.

Solving the double-banana rigidity problem: a loop-based approach

“…This is also the meaning used in robotics by (Bennett and Hollerbach, 1991) when the joint variables of a virtual contact joint are a function of the actuated robot degrees of freedom. Also, in control theory, the term passive DOF refers to non-controlled degrees of freedom which however are properly transmitted throughout the mechanism (Shiriaev et al, 2010). For example, one four-bar mechanism could have one actuated DOF, while a second four-bar mechanism on top of it could be non-actuated (passive), thus the control of the first four-bar mechanism is sought which controls both stably.…”

Generating periodic motions for the butterfly robot

“…It is being demonstrated in [5] that trajectories of the system (1) can be found by solving (18) given a set of initial conditions [θ 0 ,θ 0 ], and the constraint function (15). This is also valid for periodic motions, which are also solutions of (18) that fulfill a number of conditions.…”

“…Solving (18) V. CONTROL DESIGN Our control design techniques is based on the work of [16], [17], [18], which proposes the use of transverse coordinates for the design of feedback controllers.…”

“…with I (i) (·) being the scalar function (43)-(44) associated with the equation (18). The integral function qualifies as a transverse coordinate since it represents a robust method to quantify the distance from any given point to the desired trajectory [5].…”

Design of active disturbance rejection controller for compass-like biped walking

“…To stabilize the unstable limit cycles of underactuated walking robots, event-based control and transverse linearization were proposed. [16][17][18][19] The method of transverse coordinate control was proposed to improve underactuated robots' stability firstly by Yan et al 20 In summary, through 20 years of development, theoretically, underactuated robots' efficiency is more advanced than full-actuated robots and underactuated robots have attracted most of attention among current researchers.…”